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Singularities of functions: a global point of view

Claude Sabbah

TL;DR

This work develops a global cohomological framework for pairs $(U,f)$ with $U$ a smooth complex quasi-projective variety and $f:U\to\mathbb{A}^1$ regular, connecting Milnor-type local data to global algebraic geometry and irregular Hodge theory. It introduces global vanishing and growth-cohomology spaces, stepwise dualities, and a robust monodromy/Stokes structure that captures asymptotic behavior of $e^{-f}$-twisted objects via Stokes filtrations and Stokes data. A central pillar is the irregular Hodge theory built from Kontsevich bundles and Brieskorn lattices, yielding filtrations, dualities, and spectrum at infinity that generalize classical mixed Hodge theory to irregular settings. The framework accommodates tame cases, illuminating the spectrum at infinity, and provides concrete toric/ Laurent-polynomial illustrations with connections to mirror symmetry and exponential motivic structures. Overall, the paper unifies topological, algebraic, and analytic tools to study global monodromy, growth conditions, and irregular Hodge phenomena for regular functions on affine-to-quasi-projective varieties, with implications for arithmetic, motivic, and mirror-symmetry contexts.

Abstract

This text surveys cohomological properties of pairs $(U,f)$ consisting of a smooth complex quasi-projective variety $U$ together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function in its Milnor ball and, on the other hand, one takes advantage of the algebraicity of~$U$ and $f$ to apply technique of algebraic geometry, in particular Hodge theory. The monodromy properties are expressed by means of tools provided by the theory of linear differential equations, by mimicking the Stokes phenomenon. In the case of tame functions on smooth affine varieties, which is an algebraic analogue of that of a holomorphic function with an isolated critical point, the theory simplifies much and the formulation of the results are nicer. Examples of such tame functions are exhibited.

Singularities of functions: a global point of view

TL;DR

This work develops a global cohomological framework for pairs with a smooth complex quasi-projective variety and regular, connecting Milnor-type local data to global algebraic geometry and irregular Hodge theory. It introduces global vanishing and growth-cohomology spaces, stepwise dualities, and a robust monodromy/Stokes structure that captures asymptotic behavior of -twisted objects via Stokes filtrations and Stokes data. A central pillar is the irregular Hodge theory built from Kontsevich bundles and Brieskorn lattices, yielding filtrations, dualities, and spectrum at infinity that generalize classical mixed Hodge theory to irregular settings. The framework accommodates tame cases, illuminating the spectrum at infinity, and provides concrete toric/ Laurent-polynomial illustrations with connections to mirror symmetry and exponential motivic structures. Overall, the paper unifies topological, algebraic, and analytic tools to study global monodromy, growth conditions, and irregular Hodge phenomena for regular functions on affine-to-quasi-projective varieties, with implications for arithmetic, motivic, and mirror-symmetry contexts.

Abstract

This text surveys cohomological properties of pairs consisting of a smooth complex quasi-projective variety together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function in its Milnor ball and, on the other hand, one takes advantage of the algebraicity of~ and to apply technique of algebraic geometry, in particular Hodge theory. The monodromy properties are expressed by means of tools provided by the theory of linear differential equations, by mimicking the Stokes phenomenon. In the case of tame functions on smooth affine varieties, which is an algebraic analogue of that of a holomorphic function with an isolated critical point, the theory simplifies much and the formulation of the results are nicer. Examples of such tame functions are exhibited.
Paper Structure (22 sections, 45 theorems, 157 equations, 3 figures)

This paper contains 22 sections, 45 theorems, 157 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Topological/cohomological properties
  3. The fibration theorem
  4. Cohomological tools
  5. Cohomologies attached to a pair Uf
  6. Various expressions of the singular cohomology with growth conditions
  7. Singular homology with growth conditions
  8. Algebraic de Rham cohomology
  9. Irregular mixed Hodge theory
  10. Monodromy properties of a pair Uf
  11. Monodromy on global vanishing cycles attached to Uf
  12. The Stokes filtration attached to a pair UF
  13. The notion of a Stokes-filtered local system
  14. Pairings and Stokes matrices
  15. The twisted de Rham complex with an algebraic parameter
  16. ...and 7 more sections

Key Result

theorem 1

There exists a smallest finite set $B(f)\subset\mathbb{A}^{\!1}$, called the bifurcation set of $f$, such that the map $f^\mathrm{an}:(U\smallsetminus f^{-1}(B(f)))^\mathrm{an}\longrightarrow\mathbb{C}\smallsetminus B(f)$ is a $C^\infty$ fibration.

Figures (3)

  • Figure 1: The semi-closed disk $\widetilde{\Delta}_\rho$
  • Figure 2: The nested open subsets of the semi-closed disk
  • Figure 3: Stalks of $\widetilde{\Delta}'_c$ and $\widetilde{\Delta}_c$ at $\theta$

Theorems & Definitions (85)

  • theorem 1
  • proof : Sketch
  • lemma 1
  • proof
  • proposition 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proposition 2
  • ...and 75 more